Weak dependence of mixed moving average fields and ap- plications - PowerPoint PPT Presentation

Weak dependence of mixed moving average fields and ap- plications Based on joint work with Imma Curato and Bennet Str¨ oh | October 9, 2019 | Institute of Mathematical Finance Robert Stelzer

Page 2 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Motivation evy basis, ( A t ) t ∈ R the σ -algebra generated by 1. Let Λ be a L´ the set of random variables { Λ( B ) , B ∈ B ( S × ( −∞ , t ]) } . 2. X is called causal if X t is adapted to A t . 3. Causal MMA processes are (under moment assumptions) θ -weakly dependent. 4. Weak dependence properties are used to derive central limit theorems. 5. Aim: Generalize the concept of causality and give a suitable definition of weak dependence. Derive distributional limit theorems for such random fields.

Page 3 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Notation u is the class of bounded functions from ( R n ) u to R . ◮ F ∗ ◮ F u is the class of bounded, Lipschitz functions from ( R n ) u to R . u ∈ N ∗ F u and F ∗ = � ◮ F = � u ∈ N ∗ F ∗ u . � x 1 − y 1 � + ... + � x n − y n � , where G : R n → R . | G ( x ) − G ( y ) | ◮ Lip ( G ) = sup x � = y

Page 3 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Notation u is the class of bounded functions from ( R n ) u to R . ◮ F ∗ ◮ F u is the class of bounded, Lipschitz functions from ( R n ) u to R . u ∈ N ∗ F u and F ∗ = � ◮ F = � u ∈ N ∗ F ∗ u . � x 1 − y 1 � + ... + � x n − y n � , where G : R n → R . | G ( x ) − G ( y ) | ◮ Lip ( G ) = sup x � = y

Page 4 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Definition ( θ -weakly dependent processes) Let X = ( X t ) t ∈ R be an R n -valued stochastic process. Then, X is called θ -weakly dependent if the θ -coefficients θ ( h ) = u , v ∈ N ∗ θ u , v ( h ) − sup h →∞ 0 , → where � | Cov ( F ( X i 1 , . . . , X i u ) , G ( X j 1 , . . . , X j v )) | , F ∈ F ∗ θ u , v ( h ) = sup u , G ∈ F u , � F � ∞ Lip ( G ) � ( i 1 , . . . , i u ) ∈ R u , ( j 1 , . . . , j v ) ∈ R v , i 1 ≤ . . . i u ≤ i u + h ≤ j 1 ≤ . . . ≤ j v . Under θ -weak dependence central limit theorems can be proven under slower decay of the weak dependence coefficient compared to η -weak dependence.

Page 4 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Definition ( θ -weakly dependent processes) Let X = ( X t ) t ∈ R be an R n -valued stochastic process. Then, X is called θ -weakly dependent if the θ -coefficients θ ( h ) = u , v ∈ N ∗ θ u , v ( h ) − sup h →∞ 0 , → where � | Cov ( F ( X i 1 , . . . , X i u ) , G ( X j 1 , . . . , X j v )) | , F ∈ F ∗ θ u , v ( h ) = sup u , G ∈ F u , � F � ∞ Lip ( G ) � ( i 1 , . . . , i u ) ∈ R u , ( j 1 , . . . , j v ) ∈ R v , i 1 ≤ . . . i u ≤ i u + h ≤ j 1 ≤ . . . ≤ j v . Under θ -weak dependence central limit theorems can be proven under slower decay of the weak dependence coefficient compared to η -weak dependence.

Page 5 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Definition ( θ -weakly dependent random fields) Let X = ( X t ) t ∈ R m be an R n -valued random field. Then, X is called θ -weakly dependent if θ ( h ) = u , v ∈ N ∗ θ u , v ( h ) − sup h →∞ 0 , → where � | Cov ( F ( X Γ ) , G ( X ˜ Γ )) | θ u , v ( h ) = sup , � F � ∞ Lip ( G ) � F ∈ F ∗ , G ∈ F , Γ , ˜ Γ ⊂ R m , dist (Γ , ˜ Γ) ≥ h , | Γ | ≤ u , | ˜ Γ | ≤ v . There are no central limit theorems available achieving the weaker decay demands on the weak dependence coefficient under θ -weak dependence known from the process case.

Page 5 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Definition ( θ -weakly dependent random fields) Let X = ( X t ) t ∈ R m be an R n -valued random field. Then, X is called θ -weakly dependent if θ ( h ) = u , v ∈ N ∗ θ u , v ( h ) − sup h →∞ 0 , → where � | Cov ( F ( X Γ ) , G ( X ˜ Γ )) | θ u , v ( h ) = sup , � F � ∞ Lip ( G ) � F ∈ F ∗ , G ∈ F , Γ , ˜ Γ ⊂ R m , dist (Γ , ˜ Γ) ≥ h , | Γ | ≤ u , | ˜ Γ | ≤ v . There are no central limit theorems available achieving the weaker decay demands on the weak dependence coefficient under θ -weak dependence known from the process case.

Page 6 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Lexicographic order on R m Consider y = ( y 1 , . . . , y m ) ∈ R m and z = ( z 1 , . . . , z m ) ∈ R m . We say y < lex z if and only if y 1 < z 1 or y p < z p and y q = z q for some p ∈ { 2 , . . . , m } and q = 1 , . . . , p − 1. Define the sets V t = { s ∈ R m : s < lex t } ∪ { t } and t = V t ∩ { s ∈ R m : � t − s � ∞ ≥ h } for h > 0. V h 7 6 5 4 3 2 1 0 -1 -2 -3 -8 -7 -6 -5 -4 -3 -2 -1 0 1 Figure: V t and V h t for m = 2, h = 2 and t = ( − 2 , 4 )

Page 7 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Definition ( θ -lex-weak dependence (Curato, Stelzer and St.) ) Let X = ( X t ) t ∈ R m be an R n -valued random field. Then, X is called θ -lex-weakly dependent if θ lex X ( h ) = sup u ∈ N ∗ θ u ( h ) − h →∞ 0 , → where � | Cov ( F ( X Γ ) , G ( X j )) | θ u ( h ) = sup , � F � ∞ Lip ( G ) � F ∈ F ∗ , G ∈ F , j ∈ R m , Γ ⊂ V h j , | Γ | ≤ u .

Page 8 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Central Limit Theorem Let ( D n ) n ∈ N be a sequence of finite subsets of Z m with | D n | n →∞ | D n | = ∞ and lim lim | ∂ D n | = 0 . n →∞ Consider the random quantity 1 � X j . 1 | D n | 2 j ∈ D n What can we say about its asymptotic distribution?

Page 9 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 Central Limit Theorem (Curato, Stelzer and St.) Let X = ( X t ) t ∈ Z m be a stationary centered real-valued random field such that E [ | X t | 2 + δ ] < ∞ for some δ > 0. Assume that θ lex X ( h ) ∈ O ( h − α ) with α > m ( 1 + 1 δ ) . Let σ 2 = � k ∈ Z m E [ X 0 X k |I ] , where I is the σ -algebra of shift invariant sets. Then 1 d � X j − − − → n →∞ εσ, 1 | Γ n | 2 j ∈ D n with ε standard Gaussian, independent of σ 2 .

Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 ( A , Λ) -influenced random fields Let X = ( X t ) t ∈ R m be a random field, A = ( A t ) t ∈ R m ⊂ R m a family of Borel sets and M = { M ( B ) , B ∈ B b ( S × R m ) } a random measure. Assume X t to be measurable w.r.t. σ ( M ( B ) , B ∈ B b ( S × A t )) . Then, A is the sphere of influence and X an ( A , M ) -influenced random field . If A is translation invariant ( A t = t + A 0 ), the sphere of influence is described by the set A 0 . We call A 0 the initial sphere of influence . For m = 1 and A t = V t the above definition equals the class of causal processes.

Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 ( A , Λ) -influenced random fields Let X = ( X t ) t ∈ R m be a random field, A = ( A t ) t ∈ R m ⊂ R m a family of Borel sets and M = { M ( B ) , B ∈ B b ( S × R m ) } a random measure. Assume X t to be measurable w.r.t. σ ( M ( B ) , B ∈ B b ( S × A t )) . Then, A is the sphere of influence and X an ( A , M ) -influenced random field . If A is translation invariant ( A t = t + A 0 ), the sphere of influence is described by the set A 0 . We call A 0 the initial sphere of influence . For m = 1 and A t = V t the above definition equals the class of causal processes.

Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 ( A , Λ) -influenced random fields Let X = ( X t ) t ∈ R m be a random field, A = ( A t ) t ∈ R m ⊂ R m a family of Borel sets and M = { M ( B ) , B ∈ B b ( S × R m ) } a random measure. Assume X t to be measurable w.r.t. σ ( M ( B ) , B ∈ B b ( S × A t )) . Then, A is the sphere of influence and X an ( A , M ) -influenced random field . If A is translation invariant ( A t = t + A 0 ), the sphere of influence is described by the set A 0 . We call A 0 the initial sphere of influence . For m = 1 and A t = V t the above definition equals the class of causal processes.

Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨ oh | October 9, 2019 ( A , Λ) -influenced random fields Let X = ( X t ) t ∈ R m be a random field, A = ( A t ) t ∈ R m ⊂ R m a family of Borel sets and M = { M ( B ) , B ∈ B b ( S × R m ) } a random measure. Assume X t to be measurable w.r.t. σ ( M ( B ) , B ∈ B b ( S × A t )) . Then, A is the sphere of influence and X an ( A , M ) -influenced random field . If A is translation invariant ( A t = t + A 0 ), the sphere of influence is described by the set A 0 . We call A 0 the initial sphere of influence . For m = 1 and A t = V t the above definition equals the class of causal processes.